## Goldbach’s Conjecture

The proof of the Goldbach's Conjecture is one of the biggest still unsolved problems regarding prime numbers. Originally expressed in 1742 by the mathematician Christian Goldbach, from whom the conjecture takes its name, it was rephrased by Euler in the form in which we know it today:

Every even number greater than 2 can be expressed as a sum of two prime numbers.

In spite of the empirical evidence and the simplicity of its statement, the conjecture is resisting to every attempts of proof since almost three centuries. Several mathematicians have proved some weaker versions of the conjecture.

## Some important results

The statement phrased by Christian Goldbach is a "conjecture", so, as a matter of principle, it is a hypothesis. This means that it can be:
• True, that is all even numbers greater than 2 can be expressed as a sum of two prime numbers;
• False, that is at least one even number greater than two exists, which cannot be written as a sum of two prime numbers.
Both hypotheses are open, and scholars from all over the world are following different paths for finding a solution to what has actually become an enigma...

## Our proof strategies

Currently, there are several attempts to prove the Goldbach's conjecture, which are complete, in the sense that they come to the conclusion, but contain a series of problems, so they have not been considered valid by the international community. We instead propose some proof strategies, which are still far from being complete, but already contain interesting and unrefuted (so far) intermediate results. We hope this material will be a useful starting point for who, like us, has started looking for a proof...

## Some important results

The statement phrased by Christian Goldbach is a "conjecture", so, as a matter of principle, it is a hypothesis. This means that it can be:
• True, that is all even numbers greater than 2 can be expressed as a sum of two prime numbers;
• False, that is at least one even number greater than two exists, which cannot be written as a sum of two prime numbers.
Both hypotheses are open, and scholars from all over the world are following different paths for finding a solution to what has actually become an enigma...

## Our proof strategies

Currently, there are several attempts to prove the Goldbach's conjecture, which are complete, in the sense that they come to the conclusion, but contain a series of problems, so they have not been considered valid by the international community. We instead propose some proof strategies, which are still far from being complete, but already contain interesting and unrefuted (so far) intermediate results. We hope this material will be a useful starting point for who, like us, has started looking for a proof...

## Number theory

Goldbach's conjecture is put into the field of Number theory, the branch of Mathematics which studies the properties of integer numbers. In order to understand the proofs of the results similar to the conjecture, and very likely also to prove the conjecture itself, a sound knowledge of number theory is required. But this kind of knowledge rarely is part of a mathematician's curriculum...

## Dashed line theory

Dashed line theory is a new mathematical theory which studies the connection between the sequence of natural numbers and their divisibility relationship. Typical problems are the computation of the n-th natural number divisible by at least one of k fixed numbers, or not divisible by any of them. Due to this nature, the theory is suited for studying prime numbers with a constructive approach, inspired to the sieve of Eratosthenes...

## Number theory

Goldbach's conjecture is put into the field of Number theory, the branch of Mathematics which studies the properties of integer numbers. In order to understand the proofs of the results similar to the conjecture, and very likely also to prove the conjecture itself, a sound knowledge of number theory is required. But this kind of knowledge rarely is part of a mathematician's curriculum...

## Dashed line theory

Dashed line theory is a new mathematical theory which studies the connection between the sequence of natural numbers and their divisibility relationship. Typical problems are the computation of the n-th natural number divisible by at least one of k fixed numbers, or not divisible by any of them. Due to this nature, the theory is suited for studying prime numbers with a constructive approach, inspired to the sieve of Eratosthenes...

# Work in progress!

Our project, as it was conceived from the beginning, is very broad because it includes both a research and a disclosure part. Usually people do one thing or the other; our intent, instead, is to go on with both of them. In the last two years we have worked a lot on the disclosure part (aimed at an audience with an appropriate level of education, such as first year university students). We have completed the explanation of the proof of the Prime Number Theorem, which allowed us to introduce some concepts about number theory and to show how far the so-called "elementary" techniques can go. The writing of these articles, however, took us away from the research part, which is currently a bit sketchy. We have therefore decided, for the immediate future, to rewrite this part, organizing it better and inserting introductions that are understandable even for who is unfamiliar with dashed line theory; also the prizes page, although of quite symbolic value, will be simplified accordingly. We are therefore preparing these changes, which will be published altogether, as they are strongly linked pages. At the same time, some of our research work has not been published yet, and we wish to do it. So, in addition to a review of the existing pages, there will be also some additions. Finally, one of our research lines is still being studied in depth: in this way we hope to produce some interesting material to be published in the future.